191] NOTE OX THE EXPANSION OF THE TRUE ANOMALY. 141 
or neglecting all the terms except the first, this is 
r r c r V2r / e~ z2 dz 
= V 2ttv V r G~ r . 
Hence multiplying by ~ e r c r ^ ^ and observing that when r is large, we 
have, by a well-known formula, 
r (r + 1) = V 27tv r r c~ r , 
we obtain finally the result that when r is large the first term of A r is approximately 
ecY 
2 ' 
I take the opportunity of mentioning the following somewhat singular theorem, 
which seems to belong to a more general theory: viz. if u — e sin u = m, then we have 
log (1 — e cos u) = ^ log (1 — ae cos $), 
where 
<6 — - tan (b — to, 
-r a -r 
provided that the negative powers of a are rejected, and a is then put equal to unity. 
To show this, we have by Lagrange’s theorem, observing that 
d 
1 — ecos m) = esmmF , (l — e cos to), 
F (1 — e cos u) = F (1 — e cos m) + e - sin 2 mF\ 1 — e cos to) 
+ .j—2 sin 3 mF\ 1 — e cos to) + &c., 
and the coefficient of e r in F (l — e cos w) is 
(—Y fi r — l 
-—. ; -x <- F, cos r to -|—-— cos'' -2 to sm 2 m 
1.2...(r — 1) (r 1 
——| Fr-2 (cos r-3 to sin 3 to) + &c. [•, 
where F r = F‘ (1).